Hindawi Publishing Corporation
Mathematical Problems in Engineering
Volume 2009, Article ID 727908, 14 pages
doi:10.1155/2009/727908
Research Article
On Complementary Root Locus of
Biproper Transfer Functions
Marcelo C. M. Teixeira,1 Edvaldo Assunção,1 Rodrigo Cardim,1
Neusa A. P. Silva,2 and Erica R. M. D. Machado2
1
Department of Electrical Engineering (DEE), Faculdade de Engenharia de Ilha Solteira,
UNESP—São Paulo State University, 15385-000 Ilha Solteira, SP, Brazil
2
Department of Mathematics (DM), Faculdade de Engenharia de Ilha Solteira,
UNESP—São Paulo State University, 15385-000 Ilha Solteira, SP, Brazil
Correspondence should be addressed to Marcelo C. M. Teixeira, marcelo@dee.feis.unesp.br
Received 12 February 2009; Revised 22 May 2009; Accepted 11 August 2009
Recommended by J. Rodellar
This paper addresses the root locus locus of positive gain and the complementary root locus
locus of negative gain of biproper transfer functions transfer functions with the same number
of poles and zeros. It is shown that the root locus and complementary root locus of a biproper
transfer function can be directly obtained from the plot of a suitable strictly proper transfer
function transfer function with more poles than zeros. There exists a lack of sources on the
complementary root locus plots. The proposed procedure avoids the problems pointed out by
Eydgahi and Ghavamzadeh, is a new method to plot complementary root locus of biproper
transfer functions, and offers a better comprehension on this subject. It also extends to biproper
open-loop transfer functions, previous results about the exact plot of the complementary root locus
using only the well-known root locus rules.
Copyright q 2009 Marcelo C. M. Teixeira et al. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
1. Introduction
The analysis of the location of the closed-loop poles in the s-plane and in function of a positive
gain in the open-loop transfer function, proposed by Evans 1, 2, has found wide application
in the analysis and design of linear time-invariant systems. Recently, the root locus method
was extended to fractional-order systems 3, 4. More details about the design of controllers
based on root-locus method or depending on one parameter can be found in 5–12. The
term complementary root locus, that means locus of the closed-loop poles in function of a
negative open-loop gain, was proposed by Narendra, considering that the root locus and the
complementary root locus together form a complete algebraic curve 13.
2
Mathematical Problems in Engineering
In the design of control systems, for changing time constants, and when large loop gain
is required in inner loops, the complementary root locus is useful tool 14. Further results
about the complementary root locus can be found in 14.
There exists a lack of sources on the complementary root locus and many textbooks
about control systems present the root-locus method and its application in control design,
but do not present information about complementary root locus 15, 16. The textbooks 17–
21 present both methods.
In 14, the authors have shown that some of the steps in the method of constructing
complementary root locus construction rules are not valid for open-loop transfer functions
with the same number of poles and zeros, referred to as biproper transfer functions.
Considering this fact, this problem was studied and new definitions and analyses were
presented in 14 to plot the complementary root locus. This fact has motivated researches
to find new methods to avoid these problems and to offer a better comprehension for the
designers about this subject.
For instance, in 22 a method to plot complementary root locus using only the wellknown root-locus construction rules to plot root locus was proposed. This new technique
allows an exact plot when the sum of the number of poles and zeros is odd and an
approximated plot otherwise. An example in that paper illustrated the application of this
method to approximately plot the complementary root locus of biproper transfer functions.
In this paper an alternative procedure to plot the complementary root locus of transfer
functions that are biproper is presented. The main idea was to show that the root locus and
complementary root locus of a biproper transfer function can be directly and exactly obtained
from the plot of the root locus and complementary root locus of a suitable strictly proper
transfer function transfer function with more poles than zeros. This procedure removes the
problem cited above, because the plots are done only with strictly proper open-loop transfer
functions. The proposed method extends the results presented in 22, because it can also
allow an exact plot of the complementary root locus of biproper transfer functions, using
the well-known root-locus construction rules. An example illustrates this fact. Literature did
not present many papers about the complementary root locus, and this paper offers a better
comprehension on this subject.
2. Statement of the Problem
Given the feedback system described in Figure 1, consider that the open-loop transfer
function is biproper:
GsHs bn sn bn−1 sn−1 · · · b1 s b0
,
sn an−1 sn−1 · · · a1 s a0
2.1
where K0 , bn , bi , ai ∈ R, for i 0, . . . , n − 1 and bn /
0.
Now, define
K K0 bn .
2.2
Then, draw the root locus and complementary root locus of the closed-loop system given in
2.1 and Figure 1 for −∞ < K < ∞, using only the plots of the root locus and complementary
root locus of an equivalent system with a strictly proper open-loop transfer function.
Mathematical Problems in Engineering
Rs
−
3
Gs
K0
Y s
Hs
Figure 1: A closed-loop controlled system.
3. Root Locus and Complementary Root Locus of
Biproper Transfer Functions
In this section a solution of the problem stated in Section 2 is presented.
From 2.1, 2.2, and Figure 1, the closed-loop transfer function is
Y s
K0 Gs
Rs 1 K0 GsHs
K sn cn−1 sn−1 · · · c1 s c0
n
,
K s cn−1 sn−1 · · · c1 s c0 sn an−1 sn−1 · · · a1 s a0
3.1
where ci bi /bn , i 0, 1, . . . , n − 1.
Therefore, the characteristic equation of this feedback system is
Δs K sn cn−1 sn−1 · · · c1 s c0 sn an−1 sn−1 · · · a1 s a0
3.2
KNs Ds 0,
where Ns sn cn−1 sn−1 · · · c1 s c0 and Ds sn an−1 sn−1 · · · a1 s a0 .
Note that if K defined in 2.2 is positive, then one can directly apply the classical
root-locus construction rules to plot the root locus of 3.2. The reader can find these rules,
for instance, in 15, 16, 18–20, 23.
In the case where K < 0, Eydgahi and Ghavamzadeh 14 showed that one cannot
directly apply the classical rules to plot the complementary root locus. It is necessary to
consider additional properties, related to the roots at infinity 14. In this case, a method
presented in 22 allows an approximated plot, using only the root-locus construction rules.
The next theorem offers an alternative method to solve the problem stated in Section 2,
without the approximated plot of 22 or the additional definitions and analyses presented
in 14. It also complements the available results and offers a better understanding about this
subject.
Theorem 3.1. The root-locus and complementary root-locus plots (−∞ < K < ∞) of the system
given in Figure 1 with the characteristic equation 3.2, whose open-loop transfer function is biproper,
can be directly obtained from the root-locus and complementary root-locus plots of the strictly proper
Ns/Ds
open-loop system K
given by its characteristic equation:
Ns
Ds 0,
K
3.3
4
Mathematical Problems in Engineering
where
Ds sn an−1 sn−1 · · · a1 s a0 ,
Ns sn cn−1 sn−1 · · · c1 s c0 ,
Ns
Ns − Ds cn−1 − an−1 sn−1 cn−2 − an−2 sn−2 · · · c0 − a0 ,
K
K
.
K1
3.4
3.5
3.6
using the following expression:
The gain K can be obtained from the gain K,
K
K
1−K
.
3.7
Proof. From 3.2 and 3.5 it follows that
KNs Ds K 1Ns Ds − Ns K 1Ns − Ns
0.
3.8
Now, considering K /
− 1, then, from 3.8,
Ns −
1 Ns 0.
K1
3.9
Again from 3.8, note that KNs −Ds. So, multiplying both sides of 3.9 by K one
obtains
KNs −
K Ns
Ns −Ds − K
0,
K1
3.10
was defined in 3.6. Then, 3.3 follows from 3.10. Now, note that 3.6 and 3.7
where K
Finally, from 3.4
are equivalent. So, 3.7 can be used to calculate the gain K, given a gain K.
and 3.5, observe that the transfer Ns/Ds has more poles than zeros and so it is strictly
proper.
Remark 3.2. The proof of Theorem 3.1 can also be done using block diagram manipulations,
as described in Figure 2. Observe that in Figure 2, from 3.5, the block diagrams b and
c are equivalent. The block diagrams are very useful in the analysis and design of control
systems 15–21. A systematic procedure for reducing block diagrams and its implementation
in MATLAB program can be found in 24.
from 3.7. Note that: i for
Remark 3.3. Figure 3 describes the relation between K and K,
in the interval −∞ < K
< ∞, then K presents also values in all interval −∞ < K < ∞;
K
→ 1 . Therefore, the root-locus
→ 1− , and iii K → −∞ when K
ii K → ∞ when K
≤ 0 of 3.3 correspond to
plot 0 ≤ K < ∞ and the complementary root locus −∞ < K
the complete root loci plots −∞ < K < ∞ of the original equation 3.2. From Figure 3 one
can see that the complementary root locus −∞ < K ≤ 0 of 3.2 can be obtained from the
≤ 0 and 1 < K
< ∞ can be obtained and the root locus
root-loci plots of 3.3 for −∞ < K
< 1.
0 ≤ K < ∞ of 3.2 from the root locus of 3.3 for 0 ≤ K
Mathematical Problems in Engineering
Rs
Es
−
5
Y s
Gs
K0
Zs
Hs
a
Rs
Es
−
GsHs Ns
bn
Ds
K K0 bn
Zs
b
Rs
Es
1
K
−
Ns
Ds
Zs
c
Rs
K Ns
Ds
Es
−
Z2 s
Zs
Z1 s
K
d
Rs
Es
−
−
Z1 s
K Ns
Ds
Z2 s
Ns
Ds
Z2 s
K
e
Rs
−
K
K
1K
Z1 s
f
Figure 2: An alternative proof of Theorem 3.1: the block diagrams a, b, c, d, e, and f are
equivalent, for K / −1.
→ ∞. It means that in this situation,
Remark 3.4. Figure 3 shows also that for K → −1, |K|
Ns/Ds,
the root-loci plots of 3.3, for the open-loop transfer function K
will present at
least one asymptote, because it is a strictly proper transfer function. Therefore, at least one
branch of these plots will approach infinity in the s-plane. It was firstly presented in 14 and
the analysis above offers an alternative proof of this fact. The aforementioned gain K −1
is equivalent to the case where one has the critical gain defined in 14. Furthermore, from
→ 1, |K| → ∞.
Figure 3 note that for K
Remark 3.5. The characteristic equations 3.2 and 3.3 present the same open-loop poles,
0 it follows from these equations that Ds 0. Then, the
because for K 0 and K
procedure proposed in Theorem 3.1 preserves the open-loop poles of the original system,
given in Figure 1.
6
Mathematical Problems in Engineering
10
8
6
4
2
K
0
−1
−2
−4
−6
−8
−10
−4
−3
−2
−1
Complementary
root locus
Root
locus
Complementary root locus
0
K
1
2
3
4
from 3.7: K K/1
Figure 3: Relation between the gains K and K,
− K.
In order to exemplify the results of the above theorem, the next examples are
presented.
Example 3.6. Consider the open-loop transfer function, that is, biproper, in Figure 1:
K0 GsHs K0
s2 3s − 18
.
s2 − 4
3.11
In this case, K K0 , Ns s2 3s − 18, and Ds s2 − 4. Then, from Theorem 3.1, and 3.5,
Ns
Ns − Ds 3s − 14. Therefore, from 3.3 it follows that
Ns
Ds K3s
K
− 14 s2 − 4 0.
3.12
Ns/Ds
Note that 3.12 is equivalent to the characteristic equation 1 K
0, where
Ns/Ds
K
is a strictly proper transfer function.
< ∞ of 3.12. The open-loop poles and the
It is easy to plot the root locus 0 ≤ K
zero are p1 −2 e p2 2 and z1 4.6667, respectively. There exists one asymptote with
angle equal to 180◦ and the sections of the real axis that are occupied by this root locus are
2 ≤ Res ≤ 4.6667 and Res ≤ −2. Figure 4 shows this root-locus plot.
≤ 0 of 3.12 can be directly plotted
Now, the complementary root locus −∞ < K
Ns/Ds
using the construction rules presented in 17–21, because K
is a strictly proper
transfer function, and the problems with these rules, pointed out in 14, will not appear.
Following these construction rules one can conclude that the open-loop poles and the zero
are p1 −2 e p2 2 and z1 4.6667, respectively. There exists one asymptote with angle
equal to 0◦ , and the sections of the real axis that are occupied by this complementary root
locus are −2 ≤ Res ≤ 2 and Res ≥ 4.6667. Furthermore, the breakway points are equal to
0.4503 and 8.8830, and there is only one point s 0 where the complementary root locus
intersects the imaginary axis. Figure 5 shows this complementary root-locus plot.
Mathematical Problems in Engineering
7
4
3
2
Ims
1
0
→∞
K
0
K
0
K
→∞
K
−1
−2
−3
−4
−10
−8
−6
−4
−2
0
2
4
6
8
10
Res
Figure 4: The root locus plot of Example 3.6 from 3.12.
6
4
Ims
2
0
K
0
0 K
→ −∞
K
→ −∞
K
−2
−4
−6
−6
−4
−2
0
2
4
6
8
10
12
Res
Figure 5: The complementary root locus plot from 3.12.
Another method to plot this complementary root locus, using only the root-locus
construction rules, was proposed in 22. Following this method, define s −v. Therefore,
one can get from 3.12
N−v
K
D−v K−3v
− 14 v2 − 4 0.
3.13
it follows that
Hence, defining Kp −K
Kp v 14 v2 − 4 0.
3.14
8
Mathematical Problems in Engineering
6
4
Imv
2
Kp → ∞
Kp → ∞
Kp 0
Kp 0
0
−2
−4
−6
−12
−10
−8
−6
−4
−2
0
2
4
6
Rev
Figure 6: The root-locus plot in the plane v from 3.14.
6
4
Ims
2
0
→∞
K
0
K
0
K
→ −∞
K
→ −∞, ∞
K
−2
−4
−6
−10
−8
−6
−4
−2
0
2
4
6
8
10
Res
Figure 7: The complete root-locus plot of 3.12: root locus · − · and complementary root locus −.
Now, we can easily plot the root locus of the above equation, in the complex plane
v and with the gain 0 < Kp < ∞. Figure 6 displays this plot. Finally, from the expressions
s −v and K −Kp , one can obtain the complementary root locus in the complex plane s
and with the gain K, by rotating 180◦ the v plane around the imaginary axis 22. This plot is
the same obtained with the complementary root-locus construction rules and it is also given
in Figure 5.
< ∞ of 3.12 can be obtained from the plots
The complete root-locus plot −∞ < K
of Figures 4 and 5 and is presented in Figure 7.
Now, from Figure 3 and 3.7 one can directly plot the root locus and complementary
root locus with respect to the gain K. From Figure 3, the root locus 0 ≤ K < ∞ corresponds
< 1. The closed-loop poles for K
1, from 3.12 or Figure 4 or Figure 7,
to the region 0 ≤ K
are equal to 3 and −6. This root-locus plot is shown in Figure 8 and it was directly obtained
Mathematical Problems in Engineering
9
4
3
2
Ims
1
0
K→∞
K0
K0
K→∞
−1
−2
−3
−4
−10
−8
−6
−4
−2
0
2
4
6
8
10
Res
< 1.
Figure 8: The root-locus plot of 3.11 for K 0 ≤ K
from Figure 4 or Figure 7. Now, again from Figure 3, the complementary root locus −∞ <
≤ 0 and 1 < K
< ∞. Therefore, from Figure 7 one
K ≤ 0 corresponds to the regions −∞ < K
in Figure 7,
can get this complementary root locus, as described in Figure 9. For each gain K
the correspondent gain K in Figure 8 and Figure 9 can be obtained using the expression given
in 3.7. Finally, observe that the method proposed in 22 cannot be used to exactly plot the
complementary root locus of 3.11. Thus, this example shows that the proposed method
extends the results given in 22.
Remark 3.7. The root locus and complementary root locus plots presented in this paper, that
illustrate the proposed method, were obtained by using the software MATLAB. The main
contribution of this paper is an alternative analysis of the complementary root locus of
biproper transfer functions, without using the theorems presented in 14.
Example 3.8. Consider the open-loop transfer function K0 GsHs in Figure 1:
K0 GsHs K0
2s4 5s3 6s2 8s 12
.
3s4 8s3 9s2 12s − 16
3.15
The open-loop poles are p1 −2.4643, p2 0.6932, and p3,4 −0.4478 ± 1.7093j, and the zeros
are z1,2 −1.5750 ± 0.8068j, and z3,4 0.3250 ± 1.3455j.
Eydgahi and Ghavamzadeh 14 have shown that the complementary root locus of this
system presents a root at infinity. Then, considering that this system has the same number of
poles and zeros, the authors concluded that some rules to plot complementary root locus are
not valid in this case. In 14 this problem was studied, and new definition and construction
rules for complementary root locus were proposed. In 22 it was proposed a method and its
application to approximately plot this complementary root locus, using only the well-known
root-locus construction rules.
10
Mathematical Problems in Engineering
6
4
Ims
2
0
K → −1 K → −∞ K 0
K 0 K → −∞
K → −1
−2
−4
−6
−10
−8
−6
−4
−2
0
2
4
6
8
10
Res
≤ 0 and 1 < K
< ∞.
Figure 9: The complementary root-locus plot of 3.11 for K −∞ < K
4
3
2
Ims
1
0
−1
−2
−3
−4
−10
−8
−6
−4
−2
0
2
4
6
8
10
Res
Figure 10: The root-locus plot from 3.17 with the gain K.
Now, one can use the results of Theorem 3.1 to exactly plot this complementary root
locus. Following this procedure, from 2.1–3.2 then Ns s4 2.5s3 3s2 4s 6, Ds s4 2.6667s3 3s2 4s − 5.3333, K 2K0 /3, and from 3.5 it follows that
Ns − Ds −0.1667s3 11.3333.
Ns
3.16
Therefore, from 3.3,
Ns
Ds K
−0.1667s3 11.3333 s4 2.6667s3 3s2 4s − 5.3333 0.
K
3.17
Mathematical Problems in Engineering
11
4
3
2
Ims
1
0
−1
−2
−3
−4
−10
−8
−6
−4
−2
0
2
4
6
8
10
Res
Figure 11: The complementary root-locus plot from 3.17 with the gain K.
4
3
2
Ims
1
0
−1
−2
−3
−4
−10
−8
−6
−4
−2
0
2
4
6
8
10
Res
Figure 12: The complete root-locus plot from 3.17 with the gain K.
Ns/Ds
Note that 3.17 is equivalent to the characteristic equation 1 K
0, where
K Ns/Ds is a strictly proper transfer function. Therefore, one can plot the root locus
< ∞ and the complementary root locus −∞ < K
≤ 0 of 3.17, where the gain
0 ≤ K
using the well-known construction rules 17–21. These two plots are shown in Figures
is K,
< ∞ is given in Figure 12.
10 and 11, respectively. The complete root locus −∞ < K
Hence, from Figure 10 and 3.7, it is easy to obtain the root locus of 3.2, that
< 1 see Figure 3 for more details. This plot is
corresponds to the regions where 0 ≤ K
presented in Figure 13.
The complementary root locus of 3.2 can also be directly plotted from Figure 11,
≤ 0 and 1 < K
< ∞. Figure 14 shows this plot.
Figure 12, and 3.7 for −∞ < K
12
Mathematical Problems in Engineering
4
3
2
Ims
1
0
−1
−2
−3
−4
−10
−8
−6
−4
−2
0
2
4
6
8
10
Res
Figure 13: The root-locus plot from 3.2 with the gain K.
4
3
2
Ims
1
0
−1
−2
−3
−4
−10
−8
−6
−4
−2
0
2
4
6
8
10
Res
Figure 14: The complementary root-locus plot from 3.2, with the gain K.
4. Conclusion
A new method for plotting complementary root locus of biproper open-loop transfer
functions has been presented in this article. The main idea was to show that this plot can
be directly obtained from the root locus and complementary root locus of a suitable strictly
proper transfer function. With this method it is not necessary to perform the additional
analysis proposed in 14, that must complete the construction rules to plot complementary
root locus.
The proposed method allows an exact plot of complementary root locus, of biproper
open-loop transfer functions, using only well-known root locus rules. From the method
presented in 22, for this case, one can only obtain an approximated plot. There exists a lack
Mathematical Problems in Engineering
13
of sources about complementary root locus. This paper offers an alternative procedure and
provides a better comprehension about this subject. Finally, the educators that teach block
diagrams can also use the analysis depicted in Figure 2, as an interesting application of this
theory.
Acknowledgments
The authors gratefully acknowledge the partial financial support by FAPESP, CAPES, and
CNPq of Brazil.
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